Frequency dependent hyperpolarizabilities of polyynes

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Frequency dependent hyperpolarizabilities of polyynes

Michal Jaszunski,a) Poul J([)rgensen, and Henrik Koch^b)

Department 0/ Chemistry, Aarhus University, DK-8000 Aarhus C, Denmark Hans Agren

Institute 0/ Physics and Measurement Technology, Linkoping University, S-581 83 Linkoping, Sweden Trygve Helgaker

Department o/Chemistry, University a/Oslo, N-0315 Oslo, Norway (Received 2 December 1992; accepted 14 January 1993)

Ab initio calculations have been performed for the static and dynamic polarizability and hyperpolarizability for a series of polyynes C_2nH₂using self-consistent field (SCF) (n= 1-6) and multiconfiguration self-consistent field (MCSCF) (n=l-4) wave functions. We have considered the longitudinal component of the static hyperpolarizability and the same component of the dynamic hyperpolarizability measured in electric-field induced second harmonic generation yESHG=y( -2m;m,m,O). The frequency dependence of the polarizability and hyper- polarizability has been rationalized in terms of the coefficients in expansions in m²• The static hyperpolarizabilities vary smoothly with the chain length and satisfy y(C_2nH_{2 )}=y (C₂H_{2 )}

xn

x, where X :::-;3.0. The dynamic hyperpolarizability satisfies a similar relation where X increases slowly with m.

J. INTRODUCTION

Nonlinear optical properties of organic molecules have received increasing interest in recent years due to the po- tential use of these compounds as effective storage and switching devices. In particular, molecules with conjugated double and/or triple bonds have a large nonlinear dipole hyperpolarizability. Both ab initio and semiempirical methods have been used to calculate nonlinear dipole hyperpolarizabilities and there are numerous recent reviews and books on the subject, see for example Refs. 1, 2, 3, 4, and references therein. We here describe ab initio calculations of the static and dynamic hyperpolarizability of polyynes, which represent the simplest series of polymers.

Semiempirical methods have often been applied to study chain oligomers such as push-pull polyynes, polyenes, and polydiacetylenes because the semiempirical calculations can be performed for large enough chains to give a reliable extrapolation to the infinite polymer. Such ex- trapolations have for example been carried out for the nonlinear hyperpolarizabilities.⁵Ab initio calculations can be carried out for the smaller systems in these chains. A comparison of the semiempirical and ab initio hyperpolarizabilities indicates that significant differences can be found and that these differences may increase with improvements of the ab initio basis sets, questioning the accuracy of the semiempirical results.⁶The ab initio values for chains con- taining over 20 carbon atoms have been obtained using self-consistent field (SCF) wave functions and small basis sets. They may therefore not be reliable for the calibration

a) Permanent address: Polish Academy of Sciences, Institute of Organic Chemistry, 01-224 Warsaw, Poland.

b) Present address: UNI· C, Olof Palmes AIle 38, Dk-8200 Aarhus N,

Denmark.

of the semiempirical methods. We report in this work ab initio calculations of the nonlinear hyperpolarizability of the smallest polyynes using relatively large basis sets, considering both the effect of correlation and dispersion.

Hopefully, this provides a better calibration standard for the semiempirical calculations.

The most common approach in ab initio calculations of electric polarizabilities and hyperpolarizabilities is based on the finite field technique where the response of the system to a perturbation represented by a static electric field is computed. Most experimental data refer to optical processes and'therefore relate to frequency dependent properties. The extrapolation of the experimental results to zero frequency is difficult and it is preferable for the comparison of theory and experiment to use theoretical approaches that enable direct calculation of dynamic properties. An efficient method for accurate ab initio calculations of response properties is the multiconfiguration response function approach.⁷^,8In this approach the response of a multiconfiguration self-consistent field (MCSCF) wave function to a periodic homogeneous time dependent field is determined. The MCSCF linear response (MCLR) function first derived by Yeager and J(6rgensen determines dynamic polarizabilities.⁷In 1985, Olsen and J(6rgensen derived expressions for the linear, quadratic, and cubic response functions using a general formalism.⁸A modern implemen- tation of MCLR has been described by J(6rgensen, Jensen, and Olsen⁹ and for the quadratic response function (MCQR) by Hettema et al.¹⁰In these implementations, the linear response equations are solved using iterative methods. The response functions can therefore be calculated for MCSCF reference states with large configuration expansions and large basis sets. For SCF wave functions we have

in

a

number

of applications used the double-direct

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7230 Jaszuflski et al.: Hyperpolarizabilities of polyynes

computational scheme of Agren et al., recomputing the in- tegrals as needed both in the SCF optimization and in the response calculation. 11 This has enabled us to use larger basis sets than would otherwise have been possible since integrals are not stored.

In the next section we summarize the theoretical approach and give computational details. Section III describes the results and in the last section we give some concluding remarks.

II. THEORY AND COMPUTATIONAL ASPECTS A. Method

In multiconfiguration response calculations we solve linear response equations of the form⁹

(1) where

.El

²^]and S[2] are Hessian and metric type matrices, and

0

¹] is a gradient-type vector associated with the applied perturbation. The quadratic response function used to determine the dynamic dipole hyperpolarizability may be written in the form

«A;B,C) )"'I''''2=Nj(COI +C(2) B)7lN/(C02) +Nj(COI +C(2)

XC)7]Nf(

co I) +N~(coI) (AW +AW) X Nk(C02) -Nj(COI + C(2) (E)1~+E)~1 -colS)1,~-C02S)~I)Nf(COl)N~(C02)' (2) where

.El

^3],S[3], and

0

²^]are similar to

.El

^2],S[2], and

0

1]

but represent one higher derivative in the electronic vari- ational parameters. The A[2], B[2], and

d

2] matrices represent

0

²^]for specific perturbation operators A, B, and C.

We refer to Refs. 8 and 10 for the precise expressions and a full description of their evaluation. For the SHG dipole hyperpolarizability, the A, B, and C are dipole operators and COl

=

CO2

=

CO, the frequency of the experiment. Using the notation of Eq. (2), the xyz component of the frequency dependent hyperpolarizability may be written as

(3) where co_u

=

^-COl-co2'

In the double-direct approach for SCF wave functions the set of linear equations Eq. (1) and the double-linear transformations in the last term in Eq. (2) are calculated directly from integrals in the atomic orbital basis. SHG experiments may also provide experimental values for the second hyperpolarizability r(cocnco,co,O), where co_u=-2co, which we have evaluated from a finite field calculation of (3(cocnco,co) [see Eq. (4)].

B. Computational aspects

The ratio of the longitudinal component of the hyperpolarizability

r =

to the next largest component increases with the chain length from -10: 1 for C₆H2 to 30: I for C_SH_2•The longitudinal component is thus the most important. We study this component only and omit all tensor indices henceforth.

The polarizabilities a( -co;co) and [3( -COI-C02;COI,C(2) are calculated directly from the linear and quadratic response functions. The second hyperpolarizability r( -COl

-C02;COI,C02,0) is obtained from the first hyperpolarizability by numerical differentiation. We study only D ooh molecules for which rP=O (the superscript indicates the finite-field strength), and for any set of frequencies the differentiation is carried out as

r( -col -C02;C01>C02,0) =[3F ( -COl-C02;C01>C(2)/F. (4) This approach requires only one numerical differentiation, in contrast to standard methods where static values of

r

are obtained as the fourth derivative of the energy, the third derivative of the dipole moment, or the second derivative of the polarizability. The hyperpolarizability may be determined (see e.g. Refs. 10 and 12) from a single finite field using the field strength of 0.001 a.u. for the smaller molecules and 0.0001 for the larger ones. The numerical accuracy depends on the thresholds used for converging the wave function and the response equations. We find that the hyperpolarizabilities have errors of the order of 0.1 %- 0.5% for the chosen thresholds, while the polarizabilities are accurate to all figures quoted.

Atomic units are used throughout. For the dipole polarizability 1 a.u. = 1.648 778 X 10-41 C2 m²

r

^I ^{and for}

the hyperpolarizability13 1 a.u. =6.235378 X 10-65 C4 m4 J-3. Hyperpolarizabilities are given in 10³a.u.

C. Basis sets

Most previous calculations have been performed for the static hyperpolarizability at the SCF level. We have carried out both correlated and uncorrelated calculations of static and dynamic hyperpolarizabilities and investi- gated the basis set dependence. Table I gives information on the basis sets.

Our starting point is Dunning's correlation consistent basis aug-cc-p VTZ.14,15 It consists of a (10,5,2,1/5,2,1) GTO basis contracted to [4,3,2,1/3,2,1] and augmented with a set of primitive polarization functions, s, p, d, and!

on the carbons and s, p, and d on hydrogens. Our largest basis a is obtained by adding a set of still more diffuse functions (Is, Ip, and Id) to the carbons (geometric pro- gression of the exponents assumed). The remaining basis sets have been obtained by removing CGTOs from this basis set. Basis b is obtained by deleting one C! and one H d function (with the larger exponents). The other basis sets are derived by removing the most diffuse functions according to the scheme in Table I. An exception is the dl basis for C₆H_2,where the d orbital with exponent 0.318 is retained.

As shown in Table I, in two cases some orbitals were deleted because of linear dependencies. Using the eigenvalues of the overlap matrix as criterion, we deleted orbitals with eigenvalues smaller than 10-5.

Our basis sets differ from those used by Maroulis and Thakkar¹³ and Chopra et al. 16 In general, we use more diffuse sand p functions instead of putting more d functions on C and H atoms, partly because we consider only

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Jaszunski et a/.: Hyperpolarizabilities of polyynes 7231

TABLE I. Basis sets.

Molecule description Size SCF SCF

!label CGTO CGTO energy sum rule

CzHz (l4)a

a [6,5,4,2/4,3,2] 156b -76.85018 12.47

b [6,5,4,114,3,1] 132b -76.84755 12.28

c [6,5,4/4,3] 116 -76.84734 12.19

d [5,4,3/4,3] 96 -76.84732 12.18

C₄Hz (26)

c [6,5,4/4,3] 206 -152.54068 22.52

d [5,4,3/4,3] 166 -152.54060 22.50

e [5,4/4] 76 -152.47929 20.47

C6Hz (38)

d [5,4,3/4,3] 225c -228.24675 32.77

dl [5,4,114,3] 164 -228.19901 31.54

e [5,4/4] 110 -228.155 10 30.03

CsHz (50)

e [5,4/4] 144 -303.82725 39.47

f [4,3/3] 110 -303.82424 39.39

CIOHz (62)

e [5,4/4] 169c -379.49612 48.87

f [4,3/3] 136 -379.49571 48.81

C1zHz (74)

f [4,3/3] 162 -455.16712 58.23

"The total number of electrons is given in the parentheses.

bFive components of d function, seven components off function.

cLinear dependencies in the basis set removed from a set of 236 functions for C₆Hz and 178 functions for CIOHz.

the longitudinal component of the polarizabilities. Our longitudinal SCF polarizabilities are nevertheless similar to those obtained by these authors with their largest basis sets.

Table I gives for each basis the value of the Thomas- Reiche-Kuhn sum rule in the mixed (length-velocity) formulation.¹⁷An analysis of the corresponding values for the MCSCF functions indicates that the sum rule (number of electrons) of the active space is well described and we may expect the inactive space to be less important for polarizabilities. It is usually assumed that the sum rule is better satisfied in the length formulation than in the mixed formulation we have used.

It has been argued (see Ref. 5 and references therein) that small basis sets can be successfully used to compute the polarizabilities for long chains and polymers. Our results suggest that this may be true, but only for molecules larger than those we have studied.

D. Geometries

We have taken the geometries of the first four molecules ~nH2 (n=l-4) from Maroulis and Thakkar.¹³,lS

For C₄H2 we used the thermally averaged ^Rggeometry.

The geometries of ClOH2 and C12H 2 were optimized at the SCF level with a STO-3G basis set.¹⁹In a study of C₄H 2, the hyperpolarizabilities at Rg and Ro differ by almost

10%.13

E. The MCSCF configuration space

We use C_2vsymmetry in all calculations and describe

our wave functions using the notation (k1,k2,k3,kt/ll""/

ml, .. ./nl"")' where kj is the number of molecular orbitals of symmetry i in a given subspace. For restricted active space (RAS) SCF wave functions there are four occupied subspaces, the inactive space with all orbitals doubly occupied, RAS 1 with a restricted number of holes, RAS2 with all occupations allowed, and RAS3 with a restricted number of electrons.2o In complete active space self-consistent field (CASSCF) functions there are only two subspaces (inactive and active) and we use the notation (kl>k2,k3,k~

/ml,m2,m3,m~) as shorthand for the complete RASSCF notation (kl,k2,k3,k~0,0,0,0/ml,m2,m3,m~0,0,0,0).

The natural orbital occupation numbers obtained from second-order M011er-Plesset (MP2) calculations indicate that the most important excitations are from the bonding ^1T orbitals to the antibonding, see also Ref. 21. When analyz- ing correlation effects we therefore keep all a orbitals in- active and construct the active space from all occupied ^1T and associated

11"*

orbitals. We refer to this as the ^1T

..,.*

approximation. For C_2nH₂ (n= 1-3) we have performed CASSCF response calculations following this scheme but for C_SH2 the number of determinants becomes prohibitive.

An excellent approximation was obtained by carrying out an analogous RASSCF response calculation with at most four electrons excited from the occupied ^1Torbitals into the

1T* orbitals as described below.

We have compared the correlation energy in the ^1T--1T*

scheme with the total valence shell correlation energy estimated from a MP2 calculation with a frozen core. The

1T--1T* approximation recovers 23.5, 25.0, 27.9, and 37.7%

of this energy for n= 1-4. Thus as the chain extends this approximation gives an improved estimate of the valence correlation correction.

The number of CAS determinants increases rapidly with the size of the active space and with the number of active electrons. In some cases we have therefore used RASSCF functions instead. To verify the validity of this approach we have compared RAS hyperpolarizabilities with the corresponding CAS results, using the same correlation space with no restrictions on the occupation numbers. The results in Table II indicate that the RAS wave function reproduces the CAS correlation well and that the agreement is particularly good when four electrons are allowed to excite out of RAS1.

For the smallest system C2H2 we have performed calculations using active spaces larger than 1T--1T*. The (20001 16220/) function corresponds to full valence CAS. The computed properties do not differ significantly from the

1T--1T* results. The largest (2000/18331/) CAS contains over 2.2 X 106 determinants and is quoted as C₂H₂MCSCF for static properties in the following. For analysis of the dispersion we have selected the smaller RASB wave function, which contains

<

200 000 determinants. As shown in Table II, it gives practically the same static polarizability and hyperpolarizability.

For C4H2 we have estimated the role of the ^aorbital correlation using two RAS wave functions. The first is a (6000/3000/244010000) function with two electrons excited out of RASl, the second a (7000/2220/522010000)

function with up to four electrons excited. Essentially, the

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7232 Jaszunski et al.: Hyperpolarizabilities of polyynes TABLE II. Comparison of CASSCF and RASSCF results. a Hyperpolar-

izability in 10³a.u.

Moleculel

wave function Polarizability

~Hz

SCF 31.39

RASb 28.42

(2000/3000/3220/0000)

CAS 28.41

(2000116220/)

RASb 30.01

(2000/3000/5331/0000)

CAS 30.00

(20001183311) C6Hz

SCF 139.87

RASA^b 118.32

(13000/0330/0330/0000)

RASH' 114.48

(13000/0330/0330/0000)

CAS 113.87

(13000110660/)

'CzHz, basis b; C6Hz, basis e.

bMaximum of two holes in RAS I subspace.

cMaximum of four holes in RASI subspace.

Hyperpolarizability

3.163 2.646 2.651 3.189 3.204

65.97 55.98 58.20 58.60

first function describes excitations of up to ten electrons in a smaller active space, the second excitations of at most four electrons in a larger active space. Both RAS functions give results similar to the 1T--1T* wave function. We shall refer to the values for the second RAS function in the rest of this paper. We note that all of these functions include only part of valence shell orbitals in the active space. Fi- nally, in Table II we compare RAS and CASSCF values for C₆H2, mainly to confirm that we may use the RAS approximation for C_SH 2.

III. RESULTS

A. Static polarizabilities

Table III shows our results for the static dipole polar- izability a compared with other accurate calculations. All SCF polarizabilities have been obtained with the largest basis sets in Table I. It appears that these results are fairly close to the Hartree-Fock limit. For example, for the largest molecule for which we have used more than one basis set C_lOH 2, we find a=302.93 in the smaller basis/, so the difference is ~ 1 %. For the final basis set e we initially encountered linear dependencies in the a and 1T orbital subspaces, a further indication that an extension of the basis will not affect the polarizability.

Our results for the smaller molecules are in good agreement with other recent ab initio values. For the larger systems, we also quote the values predicted by Bodart et al.

using small basis sets.⁶

B. Hyperpolarizabilities

The SCF hyperpolarizabilities in Table IV have been obtained using the largest basis sets for each molecule. The differences between these results and those obtained with

TABLE III. Static dipole polarizabilities, a (in a.u.).

This work

~Hz

31.43

C₄Hz

86.53 C6Hz

143.16 CsHz

217.89

306.48 398.71

SCF Other ab initio

31.341,b31.42,c 30.08,d31.458,e 31.368/31.37^g 86.042,b86.36,c 76.27,d86.417^e 142.86,c130.62d 145.99i 221.33,CZ06.52d 230.59i

Beyond SCF MCTDHF This work"

30.00"

75.13"

120.58"

173.59

Other ab initio

30.23^C 30.66"

28.75^g 79.09h 86.421"

"The corresponding SCF results differ from column 2; they are ~Hz,

31.39 (basis b), C₄Hz, 86.52 (basis ^d),C₆Hz, 142.41 (basis ^d^1).

bReference 29.

"Reference 13.

dReference 16.

"Reference 22.

fReference 30.

SReference 31.

hReference 13, the SCF value for this basis set and geometry is 82.624.

ipredicted from smaller basis sets results, see Ref. 6.

smaller basis sets are larger than for a. For example, for C2H2 we obtain y=3.163 and 3.124 with basis sets band c.

Using basisfwe find 143.3 for C_SH2 and 331.8 for C_lOH 2, so the differences are of the order of 5%-15%.

The correlation effect in the 1T--1T* approximation is similar for all the molecules. The static polarizability is 10%-20% smaller than the SCF value. However, in view of the acetylene results, it is possible that a much larger active space would yield results closer to SCF.

The same applies to the static hyperpolarizabilities.

However, the differences are proportionally smaller than for aO, ~ 10%. Again, having included many active orbitals for C2H2 we find the correlated hyperpolarizability to be slightly larger than the SCF value. For larger molecules we can only estimate the valence shell correlation, it is at present impossible to include in our scheme the dynamic correlation effect.

There exist no calculations that consider correlation effects for C₆H2 and larger systems. For C₄H 2 there are two finite-field MBPT calculations but it is difficult to estimate the reliability of the computed polarizabilities as the con-

. . . d b 1322 F vergence of the MBPT expansIOn IS In ou t.' or example, the fourth-order correction to the dipole polarizability is larger than the second-order term. 22

Following other authors, we have attempted to find a simple relation between the polarizabilities and hyperpolarizabilities of different molecules. We find using a logarithmic fit to the SCF results that for C_2nH₂a=a(l)

Xn1.41 and y=y(l)xn2.^9S, where a(1) and y(1) are the

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Jaszurlski et al.: Hyperpolarizabilities of polyynes 7233

TABLE IV. Static dipole hyperpolarizabilities, r (in 10³a.u.).

SCF Beyond SCF

This Other MCSCF Other

work ab initio this work" ab initio

~Hz

3.100 3.219,b2.917c 3.204" 3.51Of 3.014,d3.143^e

C₄Hz

21.53 19.49,clS.63,d 19.1S" 24.43h 22.49^g

C6Hz

62.75 33.60,d63.4⁸ 56.29"

CgHz

175.7 116.3,d 152^g 150.0

CIOHz

377.3 C1zHz

631.2

"The corresponding SCF results differ from column 2; they are ~Hz'

3.162 (basis b), C4Hz, 21.S9 (basis ^{d), C}⁶^Hz, 62.62 (basis ^dl).

bReference IS.

"Reference 29.

dReference 16.

"Reference 30.

fReference IS, the corresponding SCF value is 3.110.

SReference 13.

hReference 13, the corresponding SCF value is 20.S3.

CzH₂values, see Figs. 1 and 2. Considering the fact that we have included six molecules with different basis sets, the agreement is very good and we believe this is characteristic of SCF limit results. The exponents are not affected significantly by correlation and we find 1.26 for a and 2.75 for y

(see Fig. 3). In this case not only the basis set but also the correlation treatment differ from one molecule to the next.

The fact that the exponent is close to the SCF value sug- gests that the same logarithmic approximation may be used for the correlated results. Finally, we note that there is no physical or theoretical basis to claim that a logarithmic expression should reproduce the exact values for the small oligomers. The exponents for both properties are in good agreement with those obtained by other authors.¹³,16 We

in au

a.

400 300 200 100

a

1 2 3 4 5 6

n in C2nH2 FIG. 1. The SCF static polarizability. The plot corresponds to a(C H)

;:::; 31.SS*n1.41, 2n Z

yin

10³^au 800

600

•

400

•

200 0

1 2

FIG. 2. The SCF static hyperpolarizability. The plot correspond to r(CznHz) =2.83*n^Z^.⁹⁸^•

also note that recent semiempirical values for various dis- ubstituted polyynes are in the range 2.9-3.3.²³

C. Frequency dependent hyperpolarizabilities

We discuss primarily calculations of the hyperpolarizability describing the electric-field induced second harmonic generation experiment

r:

^SHG⁼^{y( -} 2cu;cu,cu,0) but also report calculations describing the dc Kerr effect

Y'err

=y( -cu;co,O,O).

We performed calculations at co=O and at the nonzero frequencies 0.023 88, 0.042 82, and 0.088 56 a.u., corresponding to 1907, 1064, and 514.5 nm often used in laser experiments. We do not discuss the results for each frequency in detail. We find that our results can be well ap- proximated using the formula of Shelton

Y(CU1;C02,CU3,CU4) =y(O;O,O,O) X (1 +Acui

+

Bcu1

+ ... ),

(5) where

coi

=coj+cu~+cu~+coi and the expansion coefficients A and B are independent of the optical property. 24 We refer

yin

10³au

200

100

•

O~==---~---TI---~

1 2 3 4

FIG. 3. The correlated static hyperpolarizability. The plot corresponds to r(C_2nH₂₎=3.04*n^Z.74

•

(6)

7234 Jaszunski et al.: Hyperpolarizabilities of polyynes

to Bishop and Pipin for a discussion of the theoretical justification for Eq. (S) and related formulas. 25

We first fitted the ESHG results to determine A and B in a truncated version of Shelton's expansion

y( -2(r);(r),(r),0) =y(O;O,O,O) X (1 +6A(r)2+36B(r)4).

(6)

We report the A coefficients only since we found that B depends strongly on the basis set and the wave function.

Moreover, the dispersion in the region of interest is well described by the (r)2 term. In some cases we have performed a similar fit to the Kerr results, but we shall only discuss the ESHG/Kerr ratio of the A coefficients. Equation (S) relates different optical properties to each other, and we should obtain 2A(r)2 as the coefficient in the expansion of

~err.

It is more difficult to describe accurately dynamic polarizabilities than static. We have observed that A increases with the improvement of the basis set. For C2H2 the tabulated SCF values appear to have converged, but for C8H2 and ClOH2 the tabulated values of A are larger than those obtained using small basis sets (e.g., for ClOH2 basis

f

yields 12.90 for a and lS.1 for y). There is also a stronger dependence on correlation than for aO and yO. For example, for the three C₄H2 wave functions discussed above (the 1T-'TT* approximation and the two RAS functions) we obtain for the hyperpolarizabiIity coefficient A 11.62, 11.33, and 11.40.

Nevertheless, the general pattern of the results in Table V is clear. They demonstrate a fairly smooth dependence of the computed properties on the chain length. The coeffi- cient A increases with n for the polarizabiIity as well as the hyperpolarizability. It can be seen, in particular from the SCF results, that this increase is more pronounced for a than for y. Correlation corrections also modify the computed dispersion in a systematic way. For both properties and all molecules the correlated values of A are smaller than the SCF values. This result is not affected by the approximation used for the MCSCF function. For example, even though the static hyperpolarizability of C2H2 may be smaller or larger than in the SCF approximation de- pending on the active space, the dispersion is always weaker than for SCF.

It has been suggested that the CHF approximation may yield an incorrect description of the dispersion since a semiempirical calculation yields different relations between various optical processes. 26 We find in all our calculations (n= l-S) that the dispersion of r:SHG is about three times stronger than for ~err, in agreement with the approximate theory. The expansions in Eq. (S) applied to ESHG and Kerr yield similar values for A. The SCF ratio A(ESHG)/

A(Kerr) is 0.9S, 0.90, 0.90, 0.87, and 0.84 for n= I-S. We have computed ~err for only a few MCSCF wave functions but see no significant changes in this ratio when in- cluding the correlation effects. The same has been observed for atoms and small molecules.27,28 Correlation therefore does not invert the relation between the optical effects. The good agreement of the ESHG and Kerr results with Eq.

TABLE V. Frequency dependence of the polarizabilities and hyperpolarizabilities (a in a.u., yin 10³a.u.).

SCF MCSCF

a(O;O) A a(O;O) A

CZH2 31.39 5.17 30.01 4.88

C4Hz ^86.53 ^8.70 ^75.13 7.18

C6Hz 143.16 10.08 120.58 8.17

CgHz 217.89 11.38 173.59 8.83

CIOHz ^306.48 ^13.16

C1zHz 398.71 14.42

SCF MCSCF

y(O;O,O,O) A y(O;O,O,O) A

CzHz ^3.163 ^10.5 ^3.189 ^9.47

C4Hz ^21.89 ^13.6 ^19.18 ^11.40

C6Hz 62.75 14.0 56.29 11.99

CgHz 175.7 14.7 150.0 12.62

CIOHz ^377.3 ^15.4

C₁₂Hz ^631.2 ^15.s

(6) indicates that the results of Table V may be used to predict the hyperpolarizabilities for the third harmonic generation.

Finally, we have applied the logarithmic fit to the dynamic properties. The fit is not as good as in the static case, in particular for the largest value of (r). We find for

r:SHG(C2nH2) =r:SHG(C2H2) Xnx (7) that the optimal value of X increases slightly (S %-10% ) with (r). This occurs since the dispersion (theA coefficient) becomes larger with n.

IV. CONCLUSIONS

We have presented a systematic study of the most important, longitudinal component of the hyperpolarizability of _{C2nH 2}polyenes. There are several features that distin- guish our work from previous studies. We have extended the analysis to ClOH2 and C12H 2. The restriction to the longitudinal component has enabled us to use more typical and mostly larger basis sets. We have analyzed the role of correlation for n= 1-4. Finally and most importantly, we also describe dispersion.

It appears that correlation does not affect the hyperpolarizabilities of polyynes significantly. On the other hand, the frequency dependent values may be much larger than the static ones. In particular, for the ESHG hyperpolarizability at (r)=0.088 S6 a.u. (il.=S14.S nm) the values are 2-3 times larger than yO. The difference between the static and dynamic hyperpolarizabilities will be even more important for the THG process.

Two problems remain to be considered. The first is the question of the so-called dynamic correlation effects. Al- though not large for acetylene, it is of the same size as the valence shell correlation. Since these two correlation con- tributions differ in sign, the overall correction is rather small, and it should be verified that this holds for other polyynes. Secondly, the dependence of the hyperpolariz-

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Jaszul'iski et al.: Hyperpolarizabilities of polyynes 7235

abilities on the molecular geometry has not been considered. This may be the topic for a separate study, the quoted results for C₄H₂suggest that the geometry dependence may be important. If the geometry dependence is as strong for the larger polyynes, this may be the most significant effect neglected in our calculations.

ACKNOWLEDGMENTS

This work has been supported by the Danish Natural Research Council Grant No. 11-6844 and by (M.J.) Polish Grant No. KBN 208959101.

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